The previous article, "Barber-chemistry", considered oscillators compatible with a barberpole. Here we shall consider similar structures based on other period-2 wicks, and we shall also consider linkages that are not implied directly by any simple wicks.
1. Starting all over again, we shall now consider oscillators compatible with the following wick due to Robert Wainwright. Just as the basic element of a barberpole is the obo-spark, this wick is composed of domino sparks. Therefore we shall refer to it here as "Wainwright's domino wick", and the corresponding valency will be a "domino-valency".
With domino-radicals it is possible to construct a chemistry similar in many respects to barber-chemistry. For example, here are several oscillators consisting of pairs of monovalent domino-radicals:
Among these we see radicals that include barber-valencies as well as domino-valencies. That is, at this stage we are already dealing with diradicals which include one valency that has the nature of a barberpole, and another with the nature of a domino. An examination of P2 oscillators leads us to the conclusion that the majority can be decomposed into small parts (radicals) connected with each other by valencies of various natures.
Such decomposition is somewhat arbitrary in many cases. It is generally more convenient to define as radicals those parts of an oscillator that are least connected to other parts (separated by an interval of empty space, for example, or by a narrow "neck" of live cells in a particular generation). There should also be examples of ways to connect the remainder of the subdivided oscillator to other radicals; that is, there should be multiple radicals with valencies of the given nature.
Examples of divalent domino-radicals and some homogeneous polymerisations -- i.e., wicks and self-contained figures -- are shown below:
The domino-wick differs from a barberpole in that it has no mirror-symmetry (a barberpole has glide-reflective symmetry). Therefore domino-radicals have many more combinations of possible valency orientations. There are seven distinct varieties of divalent domino-radical, where for barber-radicals there were only three. For higher valencies this difference is even larger.
I did not attempt to deduce, for all n, the common formula for the number of varieties of n-valent domino-radicals. There are several reasons for this. First, the problem is much more difficult than its equivalent for symmetric linkages. Second, any such formula will be true only for covalent linkages -- and later we shall consider polar linkages as well. Third and most important, multivalent radicals with linkages of a single type do not play a particularly important role in the structure of oscillators. Radicals having valencies of multiple types are more important -- simply because such radicals are smaller, as a rule. And taking into account all possible varieties of multi-type linkages is impossible.
Of the seven types of divalent domino-radicals, three are very similar to three types of divalent barber-radicals. In them, one linkage can be mapped onto the other by a 90- or 180-degree rotation, or a translation (for 0-degree "rotations"). These can be considered valencies of types I, L and U -- everything stated about barber-radicals of these types also holds for domino-radicals
The remaining four linkage types for divalent domino-radicals involve a reflection across an orthogonal or diagonal axis, along with an associated translation. Some of these 'reflective linkage types' are reminiscent of type-L radicals, but they cannot form homogeneous self-contained figures. The only homogeneous polymerisations of reflective radicals are wicks. However, it is possible to create inhomogeneous self-contained oscillators using a combination of radicals of different reflective types (see domino.lif, item 5).
Some divalent domino-radicals also include barber-valencies closed by preblocks, so they are really radicals of a higher valency, but with some linkages of a different nature. Quadrivalent variants that can yield agars are most interesting; some of these are shown in an associated file (items 2, 3, and 6 in domino.lif).
Quadrivalent radicals are also possible where all four valencies are domino-type. Of these, the following radical is most interesting. In the densest polymerisation the agar similar to Don Woods' 'squaredance' is formed. In the absence of another name, we shall refer to it as his rock-and-roll agar (in my opinion rock-and-roll is more "square" than a squaredance -- a common rock-and-roll chord sequence is referred to as "square", or "quadrate", by Russian rock musicians). By the way, can anyone provide a stabilization of this agar perimeter for larger squares?
2. Another wick of Robert Wainwright's also consists of lines of domino sparks in one phase. But the other phase of the wick consists of obo sparks, similar to a barberpole. I shall name his 'obo-do wick'. Here "do" is simultaneously the initial syllable of "domino" and an abbreviation of the words "double o", corresponding to the RLE encoding of a domino spark.
Obo-do linkage is similar in many respects to domino linkage. It also is nonsymmetric, and thus yields 7 types of divalent obo-do radicals.
Among the quadrivalent obo-do radicals the following radical is most interesting. Its polymerisation also yields a rock-and-roll agar. Thus, this agar has a dual nature: it can either be decomposed into cruciform radicals connected by domino-linkages, or alternatively into square radicals connected by obo-do linkages. However, sparser agars formed from obo-do radicals differ from their domino analogs.
3. The following wick, found by Dean Hickerson, consists of oblique triplets. Triplets are connected with each other by a single live cell; therefore we shall use the term "dot valency" to refer to the valency connecting triplets with each other.
The domino part of one triplet can be linked with the single dot of the next triplet in two mirror-image orientations, so these radicals can be connected without worrying about chirality (left- or right-handedness). So for dot-valency, as for barber-valency, there are only three types of diradicals instead of seven: U, L and I. However, in contrast to barber-valency, the direction of linkage is important: dot linkage has polarity.
We shall say that the dot extremity of an oblique triplet has positive polarity, and the domino extremity is negative. The polarity of a dot linkage reverses every generation. We shall name this type of linkage a 'semipolar' linkage to distinguish it from a 'full polar' linkage in which polarity does not vary with time.
With a semipolar symmetric linkage of this type, there are six types of diradicals instead of three: for each divalent type, variants may have valencies with matching polarity or opposite polarity. Since the polarities change in synchrony with each other, the type of the diradical does not vary with time -- so the type can safely be used to predict the behavior of the radical in homogeneous polymerisations.
For example, type-I diradicals with valencies of opposite polarity can be connected immediately to copies of themselves, forming wicks. But type-I diradicals with valencies of matching polarity cannot be connected in this way; successive radicals must be in opposite phases for the connections to work. By contrast, alternation of phases is not generally necessary for barber- and domino-radicals (I did it sometimes to derive a tighter joint).
There are some small divalent dot radicals that can form wicks that are just as simple as a barberpole, domino or obo-do-wick. These wicks are connected by the same kind of dot-valency as Dean Hickerson's triplet wick. Thus their basic components can be classified as type-I dot-valency diradicals with valencies of opposite polarities, just like the oblique triplet. These wicks also share other characteristics of oblique-triplet wicks.
We shall name one of these wicks (due to Robert Wainwright) the 'dash-and-dot wick' because of its appearance. The valency connecting this wick is the same familiar dot-valency discussed above, and almost all dot radicals found for a triplet wick will work perfectly with the dash-and-dot wick.
In the associated files of search-program results (triplet.lif and dashdot.lif) many radicals will appear to consist of smaller elements. I did not discard such composite radicals, as they show possible methods of constructing oscillators from smaller elements.
Because of the semipolarity of dot linkage there are, for example, 4 subtypes of type-X quadrivalent radicals. The smallest such radical looks like a diagonal two-bit spark: two opposing valencies have positive polarity, and the other two are negative. The densest polymerisation of this radical is the known agar consisting of diagonal two-bit sparks. A sparser variant is also shown below.
Segments of a barberpole or obo-do wick of any length are also quadrivalent dot radicals. In particular, the domino-spark is also a quadrivalent dot radical. Therefore, for example, the rock-and-roll agar can be considered to be a polymerisation of domino-spark radicals -- a much smaller basic unit than the radicals considered earlier, which consist of four dominoes.
Valencies can be identified in other period-2 wicks without any great difficulty. Next we shall consider another mode of detection of valencies, starting with any simple oscillator rather than a wick.
4. We shall consider, for example, a beacon. A beacon consist of two "poorly connected" halves (in one phase they make separate islands, and in the other they are connected at only one point). Can we find other "beacon-radicals" -- that is,. can we find oscillators consisting of a half-beacon connected via a blinking diagonal spark to some other structure (different from a half-beacon)? Here are examples of such oscillators:
And there are also examples of divalent beacon-radicals, though unfortunately the majority of them can not form homogeneous wicks:
Such a wick can be constructed from the following (known) divalent beacon-oscillator, which is simultaneously a divalent barber-oscillator -- it is based on a quadrivalent radical with two barber-valencies and two beacon-valencies. This also allows the construction of an agar:
5. A toad is another simple oscillator that breaks up into two islands in one of its phases. Here are some examples of divalent oscillators having toad-valency, and also wicks based on this type of linkage:
6. The following pattern, sent to me by Heinrich Koenig, is a perfect example of a series of complex constructions using the linkage types mentioned above, along with other period-2 valencies.
Nicolay Beluchenko, April, 2004.